The Mathematics of George Washington


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I recently learned some things about how the young George Washington did math, including surveying. Mathematician and historian V. Frederick  Rickey gave a talk 2 nights ago at the Mathematical Association of America here in DC, based on his study of GW’s “cypher books”, and I’d like to share a few things I learned.

(1) The young George appears to have used no trigonometry at all when finding areas of plots of land that he surveyed. Instead, he would ‘plat’ it very carefully, on paper, making an accurate scale drawing with the correct angles and lengths, and then would divide it up into triangles on the paper. To find the areas of those triangles, he would use some sort of a right-angle device, found and drew the altitude, and then multiplied half the base times the height (or altitude). No law of cosines or sines as we teach students today.

(2) He was given formulas for the volumes of spheroids and barrels, apparently without any derivation or justification that they were correct, to hold so many gallons of wine or of beer. (You probably wouldn’t guess that you had to leave extra room for the ‘head’ on the beer.) Rickey has not found the original source for those formulas, but using calculus and the identity pi = 22/7, he showed that they were absolutely correct.

(3) GW was a very early adopter of decimals in America.

(4 ) This last one puzzled me quite a bit. It’s supposed to be a protractor, but it only gives approximations to those angles. The results are within 1 degree, which I guess might be OK for some uses. I used the law of cosines to convince myself that they were almost all a little off. Here’s an accurate diagram, with angle measurements, that I made with Geometer’s Sketchpad.

His method was to lay out on paper a segment 60 units long (OB) and then to construct a sixth-of-a-circle with center B, passing through O and G (in green). Then he drew five more arcs, each with its center at O, going through the poitns marked as 10, 20, 30, 40, and 50 units from O. The claim is then that angle ABO would be 10 degrees. It’s not. It’s only 9.56 degrees.

Telescope Fix, Scudding Clouds Over Moon, Tom Turkeys Hiding, and Successful Cloud Chamber!


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I had a productive 24 hours!

  • Night before last, I think I finally got Sky Wizard Digital Setting Circles installed on the 14″ alt-az telescope we were most generously donated by Alan Bromborsky. (That’s me, in the operations cabin  at Hopewell Observatory, taking a break and a picture, long before completion.)img_6169
  • So I went out to look at the sky at 1 AM. I saw no stars, but the 80% gibbous moon appeared to race dramatically through the clouds
  •  That afternoon, as I was driving out, I saw 5, maybe 6 tom turkeys playing hide-and-seek with me behind the trees. Believe me, they are REALLY GOOD at hiding behind little saplings, logs, and rocks! Or if you don’t believe me, ask anyone who’s tried to hunt them.
  •  Late that evening, I got a dry-ice-and-isopropanol particle detector working for the first time. (I had tried and failed, when I was a teenager, some 50 years ago, and failed several other times since then as well.) If you look at my little video, you can see the particles more easily than I could with your naked eye as I was filming it. Don’t ask me yet which ones are muons, which are alpha particles, and which are beta particles, because I don’t know yet. But you could look it up!

Productive 24 hours!

A Talk at AHSP on Telescope Making


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Two weeks ago was the Almost Heaven Star Party on the slopes of Spruce Knob, West Virginia, sponsored and organized by the Northern Virginia Astronomy Club (NOVAC). The weather was wonderful, and we could see the Milky Way and lots of Messier objects with our naked eyes, every single night for four nights. This is by far the longest stretch of good weather I’ve ever experienced up there at The Mountain Institute.

(Friday and Saturday, it was only clear for a few hours, but Sunday and Monday nights were clear all night, AND there was NO DEW to speak of!! Wow!!)

During the daytime, there were lots of talks and also activities and expeditions such as hiking, spelunking, visiting the National Radio Astronomy Observatory at Green Bank, canoeing, and Phun With Physics and arts & crafts for kids. I particularly enjoyed the talks on Russell Porter (the founder of amateur telescope making in the US and one of the major designers of the 200-inch telescope at Palomar), LIGO (detection of gravity waves), and Rod Molisse’s talk on 50 years of mostly-commercial telescopes as seen in the pages of various astronomy regime.

I was one of the speakers and gave a little talk on telescope-making. If you care to sit through it, you can find it along with all of the other talks (many of which I missed for various reasons) at this web-page.

I brought my home-made 12.5″ Dob-Newt [shown to the left in the picture below] and added about a dozen items to my formal list of Messier objects. (I had already seen all 100+ objects, but hadn’t recorded enough details on them to be able to earn one of the ‘merit badge’ pins from the Astronomical League, so I’m going through the list again


(If you didn’t know: Charles Messier loved hunting comets about 220 years ago, with what we would consider today to be a fairly small (4″ diameter) refractor that he used from downtown Paris, not far from where I lived back in 1959. Comets look like fuzzy patches in the sky, and so do galaxies, star clusters, and illuminated clouds of gas, all of which are MUCH farther away and MUCH larger. Comets are part of our own solar system, and move noticeably from one night to the next against the apparently fixed background of stars. Messier is credited with discovering 13 comets. But when he discovered a fuzzy item in the sky that did NOT move, he would record its location and its appearance, so as to avoid looking at it again. He published and updated this list a few times before he died, 199 years ago. Nowadays, his list of things-to-be-avoided are some of the most amazing and beautiful things you can see in the night sky. I’ve tried imaging a few of them, but am very, very far from being proficient at it. I attach my best one so far, of something called the Dumbbell Nebula. No, it’s not named after me. And thanks to Mike Laugherty for helping with the color balance!)


Sex and Math: The Zero to Two Percent of Our Actual Lives That Rules Our Lives


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Someone (probably Louis CK) pointed out that even though we all obsess about sex (and being attractive to the persons with whom we long to have sex), it’s actually a very, very tiny part of one’s actual (as opposed to our wishful) life.
[Of course, any species that survives needs to have a strong drive for procreation, or they won’t leave enough offspring behind to carry on. (Just ask pandas what happens when they seemingly couldn’t care less… They are, if you hadn’t noticed, nearly extinct in the wild. Rats, rabbits, ants and cockroaches survive by being unbelievably fertile — and sneaky.]
Let’s try to do some math on this.
Have you ever tried figuring out what percent of your entire life has consisted of you actually making love (having sex) with another person?
I am not counting masturbation here because
  • I don’t trust any data on this, even Masters and Johnson, so any guesses on my part would be just that
  • I’m only talking about having sex with another person, either Penis-In-Vagina or any other sort of sexual activity, which I am not about to list here —  use your own imagination if you want to. 
  • And no, I’m not going to tell you anything about my own habits or those of anybody else I know. However, if you want to tally up your OWN time doing ‘that’, feel free.
Let’s look at the high end of the spectrum of those having lots of sex first.
My guess is, based on observations of personal experience and what I’ve observed with people I knew well:  that only with the very horniest newly-weds or with a couple who have just entered a super-sensual, brand-new sexual relationship,  time would a couple be spending, say, as much as four hours a day actually’doing it’. Why? For one thing, sex is exhausting. Also, the tissues involved are delicate and can only take so much rubbing, no matter how well lubricated they might be. Plus the couple need to sleep, eat, wash, and most likely do something productive like going to school or work.
That very high figure for someone getting a HUGE amount of nookie is 4 hours  out of 24 hours in a day, or 1/6, which is about 17% of the time. Let me repeat: that’s extraordinarily high, and from my own observations (thin walls allow one to hear… and so on) only seems to last for the first few weeks, because then they get sore, exhausted, sated, and somewhat jaded.
After that, they’d be lucky to be ‘doing it’ for an hour or two a day, which is between about 4 and 8% of the time.
But let’s compare that to the rest of their lives for this remarkably horny and lucky couple… Let’s suppose that they are 20 years old, and let’s suppose that they each had sex a few times in high school and after (college, military, working, whatever, and I’m making no assumptions about whom they are doing this with). Maybe they got laid 2 – 3 times a week from age 16, at about an hour each time (girls, feel free to scoff at my suggestion that a typical male teenager can actually last that long) plus, now that they are not living at parents’ home any more, they have had several tumultuous, sexy relationships one after the next, each time spending an average of 2 hours actively ‘doing it’ each 24-hour period, for the last two full years.
So let’s make that say 6 hours a week for 2 years plus 28 hours a week for the last 2 years – and these people are those who are towards the very far right hand end of the frequency curve distribution. I don’t know whether the distribution of ‘nookie-hours’ if graphed, is ‘normal’ or ‘skewed’one way or the other. But in any case, most people ‘get’ a lot less sex than this. In fact, when I was in high school and college, the vast majority of my (male and female) friends my age or younger were virgins. No, I’m not going to tell you what age or to whom I lost my virginity. I will keep my fond memories to myself, and hope you will do likewise.
Here’s the arithmetic:
Denominator (total hours lived) for this very lucky pair of 20-year olds: 24 hours per day time 365. days per year times 20 years old equals 175,200 hours total that they have lived so far. (D)
Numerator ( total hours having sex with another person since they were born) Assuming this very sexually active couple who have sex 6 hours a week times 104 weeks (two years from ages 16 – 18) plus 28 hours a week times 104 weeks (two more years, aged 18 – 20) equals 3,536 hours having sex, grand total, (N)
I used a calculator to divide N by D and I get about .02, or two percent of their life so far. That’s tops.Two percent of their life.
I suppose it is theoretically possible that a married couple of 60 years, at age 75, has been actively having sex a full hour every single day of their married life (since age 15). Extremely rare, I know… But for this tiny handful of  extraordinarily horny, happy, healthy and lucky people, I get a grand total of 1/30 of their life, or a tad more than 3%. And I bet that you could probably hold a nice party for every single one of these lucky 75-year-old living couples — from all over the world — in the cafeteria of the elementary school nearest my house. Not sure how much standing room there would be, but this is out of our current world population of over 6 billion people. So they are outliers of outliers of outliers.
The vast majority of people would have much, much less sex than that, I predict with confidence.
Additionally: a good number of people die before losing their virginity. I have no idea what percentage do, but during the old days, it was probably well over half  of them, infant mortality rates being what they were — and I’m assuming that most of those babies and children who died at very early ages from disease, murder or accidents weren’t being sexually abused during their short, sad lives… Many soldiers die in combat having had sex perhaps 0 to 6 times in their entire lives, if I can trust various memoirs I’ve read by veterans of various wars…
So that’s the far left hand end of the ‘bell-shaped’ curve: 0%.
While there are a tiny handful or two-to-three percenters out there, I would bet (if we could test it somehow) that over 99.99% of the population has sex, on average, between ZERO and TWO percent of our entire lives.
But, oh, how we obsess over ‘it’!!

When was the last time you spent a night under the stars?


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If it’s been a while since you spent time looking up at the heavens with your naked eyes, binoculars and telescopes, looking at planets, stars and galaxies, then this Saturday might be your night.

The Hopewell Observatory is having an open house on Saturday, July 2, 2016, and we have a variety of scopes to look through. Some of the scopes will be under our roll-off roof and some will be rolled out onto the small lawn outside the observatory itself.

Mars, Jupiter and Saturn will be very conveniently placed for viewing right at sundown, and if it’s dry and clear enough, we should be able to see the Milky Way. Many nebulae, open and globular clusters, galaxies, and double or triple stars will be visible as well.

You are invited!  And it’s free!

The location is about an hour due west of Washington DC by way of I-66, near the town of Haymarket, VA. For detailed directions, follow this link, which I posted for one of the dates which got canceled because of bad weather. Ignore the date, but do pay attention to the fact that we have no running water! We have bottled water and a composting toilet and hand sanitizer. Plus makings for coffee, tea, and hot chocolate – all gratis.

picture of hopewell

The picture above is of one of our telescope mounts, which carries several telescopes and was set up to take astrophotographs at the time. Below is a picture of the outside of the observatory shortly after a snowstorm.. Notice that there is no dome – instead, the galvanized steel roof rolls back on the rails and columns to the right of the picture when the scopes are in use.


If you have your own telescope, feel free to bring it. If it needs electricity, we have an outdoor 120VAC outlet, but you should bring your own extension cord and plug strip.  If you want to stay all night, that will be fine, too! If you feel like bringing a cot or a tarpaulin and a sleeping bag, that’s equally OK by us! Show up at or near sunset, and stay until the sun comes up, if you like!

Warning: the area definitely has insects, such as ticks and chiggers, which appear to avoid everybody else and to do their best to attack me. I strongly recommend long pants, shoes/boots, and socks that you can tuck the pants into. Tuck your shirt into your pants as well, and use bug spray, too. I have personally seen plenty of deer, cicadas, moths, wild turkeys, squirrels, and birds, and I have heard from a neighbor that a bear tried to eat his chickens, but other than the insect pests, the wildlife stays out of your way.

Again – for detailed directions, look at this link.

Any Moth Experts Out There?


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We found these two beautiful moths that flew into the operations cabin at the Hopewell Observatory a couple of nights ago, and we have no idea what type they are. Never seen them before and can’t find any images identical to them. (One species is similar, though.)

Any suggestions will be welcome.

Ain’t they purty li’l things?

And when they opened their wings they were even more spectacular, but I didn’t get a good shot.BTW the yellow-and=red moth is sitting on the struts of a telescope made by Alan Bromborski.

Movie about Ramanujan


I recommend going to see ‘The Man Who Knew Infinity’, about the famous Indian mathematician Srinivasa Ramanujan, whose originality astonished the greatest mathematicians of his day, G H Hardy and Littlewood.

It’s a historical romance rather than a documentary, so there are parts that are completely made up — like German Zeppelins bombing Cambridge University during World War 1, and off-duty British soldiers beating up Ramanujan for being a foreign draft dodger. A weird aspect is that the actors and directors don’t seem to be on the same page about the correct pronunciation of Ramanujan’s name. They mostly put the accent on the third syllable, where as far as I can tell it should be more like ruh – MAHN – uh – jahn.

However, there was such a bombing near-by during the war, and the British army did have temporary hospitals for wounded soldiers during that war, and Ramanujan did contract tuberculosis and die at the sadly early age of 32, after he had gone back home. And apparently his mother had indeed been intercepting letters to and from his wife, so they were completely cut off from each other, as the movie states.

And of course, they can’t really explain much of the mathematics – heck, I can’t understand much of anything that Ramanujan did! Even his simplest formulas are way, way over my head! Dramatizing partitions of a number was probably a wise move, since it’s one of the few things that an ordinary person might understand.





Evolution is SLOW

A long time ago I had a course by Prof Jim Sandeful of Georgetown U and Dr Monica Neagoy on teaching with “discrete math” — very useful and interesting stuff that often does not get discussed in the standard American curriculum. I enjoyed it a lot.

Among other topics, I decided to write a little computer program that would model exponential random decay of radioactive elements. (Iirc I did this in Pascal and in BASIC, on the C-64, IBM-PC, Apple II, and Commodore Amiga. That was fun.)

One subtopic that came up, but which I never figured out how to model, was how to describe the frequency of some trait (eg red hair, striped tail, or growing a third eye…) in a population. I had long thought about how to do that but not until today did I begin to make some progress, so please allow me to share.

I’m going to make up a very-much simplified example using a Punnett square, something like this:

If you have no idea what this means, let this non-biologist try to explain as best I can:

Upper-case B and lower-case b in this diagram stand for two different versions of a particular (but mostly imaginary) gene that controls whether a person has blue eyes or brown. In this hypothetica example, the upper-case B gene causes brown eyes and is dominant, where the lower-case “b” causes eyes to be blue and is recessive. Thanks to the magic of sexual reproduction, you get two copies of each gene, 1 from Mom and one from Dad, whether they stick around and raise you or not. (You have two similar-but-not-identical copies of each chromosome except for the X and Y chromosomes; your two versions of each gene are carries in corresponding locations on each of the two chromosomes. If I got this right.)

If you have brown eyes, then your genes might be BB or they might be Bb or bB (same thing). If you have blue eyes, then you have bb genes for sure — again, in this hypothetical scenario.

This Punnett square shows the probability of what will happe if two parents who carry Bb genes have sex and produce offspring. It reminds me very much of how we use an area model to show that (X + Y)*(X + Y) equals X^2 + 2*X*Y + Y^2.

In any case, each of those parents carries Bb genes, and when the eggs and the sperm cells are manufactured inside the parent’s ovaries and testes, one or the other version of the gene is put inside, but not both. And it’s random. So since each parent has a Bb gene, its probability of passing along upper case B (brown) is 1/2 or 50%, as is the probability of passing along lower case b (blue eyes).

You can now find the probability of all of the outcomes shown in the interior of the diagram. The upper left hand corner is BB, pure brown eyes, with probability 1/4 because 1/2*1/2=1/4 and also in this case all of the sections really do have equal areas.

The upper right hand and lower left hand corners represent the Bb cross; the child will have brown eyes. The probability of a Bb cross is 1/4 plus 1/4, or 1/2.

The lower right hand corner is the region representing the probability of pure bb offspring  which have (recessive) blue eyes. The probability of bb is 1/4.

Now let us add a couple of features.

1. This is not just a single mom-dad pairing: this is a representation of an entire reproducing population where genes B and b are present, each 50% of the time.

2. Let us also pretend that the bb combination is fatal: not a single one of them survive to adulthood and to leave offspring. (This is a very extreme hypothetical example of how evolution operates. Normally Deleterious genes aren’t so uniformly fatal!) or alternatively, a breeder of plants or animals might decide to not permit any of the blue-eyed bb offspring to reproduce. Eugenicists used to advocate sterilizing anyone who exhibited harmful, recessive genes,in order to improve the remainder of the human race.

At first glance, You would think that this sort of genetic selection, either by artificial or natural means, would work very quickly, and that after just a few generations, the proportion of the population that was blue-eyed would vanish.

Jim Sandefur said no, it would take a really long time. I forgot the details, and just worked them out today. I’ll work out the details for you later when I have a larger screen. But:

Bottom line: even with this 100% culling of recessive genes, the proportion of blue eyes goes down as the harmonic series (1/X), where X is 4, then 5, 6, 7, 8, etc

So if the first generation has 1/4 (25%) blue eyes, and if every single individual with blue eyes is somehow prevented from reproducing, then the next generation will still carry the lower-case b gene 1/5 (20%) of the time.

And if children with blue eyes (bb) are still prevented from reproducing, the third generation will still pass on  the lower-case b gene one-sixth, or 16.67% of the time, and the next generation will pass on the lower-case b gene one-seventh (14.29%) of the time. The next generation passes on b genes one-eighth (12.50%) of the time, then one-ninth of the time (11.11%), then one-tenth of the time (10.00%) and so on. At first the decrease is pretty rapid, but after that it slows to a craw, and the world would never be entirely free of the pure bb. After 100 generations, there still would be 1/103 (almost 1%) of the population carrying genes that can pass on blue eyes.

At 25-30 years for a human population to reproduce, you are talking about 2,500 to 3,000 years!

However, the fraction of the population that actually is born with blue eyes apparent to everybody will fall much faster.  The proportions would be 1/4 in the initial generation, followed by 1/9, then 1/16, then 1/25, then 1/36, then 1/49, then 1/64, and so on, with the ratio being 1/X^2 (one over x-squared) rather than 1/x.

So, by 10 generations, under this hypothetical, 100%-effective sterilization or extermination regime, the proportion of the population with visible blue eyes would have fallen to 1/169, about six-tenths of a percent. However, the fraction of the population that still carries the genes for blue eyes would remain at 1/13 of the population, about 7.7% of the total.

However, perhaps conditions might flip-flop. In my hypothetical problem here, perhaps the conditions making blue eyes fatal would disappear after a number of generations. (Even if Hitler’s nasty 1000-year Reich would not have been enough to eradicate whatever enemy genes!) In fact, perhaps the reverse would be true: having brown eyes would be a fatal handicap under some conditions. Then the prevalence of blue eyes would rise to the fore in their place, but there would be an enormous die-off of all those who had brown eyes, which would mean the vast majority of the population. So all that would be left would be those formerly recessive genes, and the formerly dominant genes would be wiped out completely.

More realistically: recessive genes that make people susceptible to die from some particular disease or parasite or environmental factor do definitely get reduced in frequency over time, as I hope I have shown. However, they do not disappear completely for a very long, long time (if ever!) unless the entire population is reduced to just a handful of individuals, none of whom carry that gene, just by chance.

Evolution does work on those time scales. Human societies and any proposed eugenics program do not. Evolution has no direction, and is essentially blind, like a mathematical algorithm.

People often say that everything happens for a reason. Often, that reason is simply the laws of probability, which are extremely hard for most people to handle. Myself included.